Distributed Predefined-Time Algorithm for System of Linear Equations Over Directed Networks

In this brief, we propose a predefined-time algorithm based on multi-agent networks for solving linear equations of the form
$Ax = b$
, where the predefined-time can be predesigned by the user and each agent only knows partial rows of the augmented matrix
$(A\; b)$
. By reformulating the system of linear equations as an unconstrained optimization problem and applying the time-base generator technology to optimization approaches, we provide a gradient-based predefined-time algorithm over weight-balanced directed graphs, which can converge to a solution at a given time. Finally, this theoretical results are illustrated by a numerical examples.